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Flexure

Eurocode 2 offers various methods for determining the stress-strain relationship of concrete in flexure. For simplicity and familiarity the method presented here is the simplified rectangular stress block (which is similar to that found in BS 8110). In determining the resistance of sections, the following assumptions are made.

Plane sections remain plane.

Strain in the bonded reinforcement, whether in tension or compression, is the same as that in the surrounding concrete.

Tensile strength of the concrete is ignored.

Stress distribution in the section is as shown in Figure 2 - simplified rectangular stress block

Limiting neutral axis depth

As a beam experiences more moment, it is often considered good practice to limit the depth of the neutral axis to avoid ‘over-reinforcement’ (i.e. to ensure that the reinforcement is still behaving elastically at failure). This is not a Eurocode 2 requirement and is not accepted by all engineers. A limiting value for K can be calculated (denoted K’) as follows.

Beyond K’ compression reinforcement is required. Neutral axis depth x varies with the amount of redistribution of moments. K’ = 0.207 is valid where there is no redistribution but for redistributions of 15%, K’ = 0.168 is appropriate. In the UK it is often recommended that K’ should be limited to 0.168 to ensure ductile failure.

Proof: compression reinforcement

In some cases, compression reinforcement is added to increase section strength where section dimensions are restricted i.e. where K > K’. It may also be added to reduce long term deflection or to decrease curvature/deformation at ultimate limit state. It may also be added to reduce long term deflection or to decrease curvature/deformation at ultimate limit state.

With respect to Figure 1, we now need to consider an extra force Fsc = 0.87As2 fyk

The area of tension reinforcement can now be considered in two parts, the first part to balance the compressive force in the concrete, the second part is to balance the force in the compression steel. The area of reinforcement required is therefore:

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